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Axioms 2017, 6(2), 7; doi:10.3390/axioms6020007

Fourier Series for Singular Measures

1
Department of Mathematics, Butler University, Indianapolis, IN 46208, USA
2
Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, USA
*
Author to whom correspondence should be addressed.
Academic Editors: Palle E.T. Jorgensen and Azita Mayeli
Received: 21 February 2017 / Revised: 23 March 2017 / Accepted: 24 March 2017 / Published: 28 March 2017
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
View Full-Text   |   Download PDF [276 KB, uploaded 28 March 2017]

Abstract

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f L 2 ( μ ) possesses a Fourier series of the form f ( x ) = n = 0 c n e 2 π i n x . We show that the coefficients c n can be computed in terms of the quantities f ^ ( n ) = 0 1 f ( x ) e 2 π i n x d μ ( x ) . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ -bandlimited. View Full-Text
Keywords: Fourier series; singular measure; Kaczmarz algorithm Fourier series; singular measure; Kaczmarz algorithm
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Herr, J.E.; Weber, E.S. Fourier Series for Singular Measures. Axioms 2017, 6, 7.

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